This is another shot from the quantum superposition viewer. Â It’s neat as op art, but I’m still trying to fully grasp the realization I made that this represents an unchanging distribution of a unit of rotational motion. Â The state is fixed, but it contains within it an orbiting particle.
I’m playing around with quantum mechanicalÂ superpositionÂ and hydrogen wave functions again. Â This time with really simple 2 dimensional renderings. Â I’ve been trying to get a grip on some of the basic principles and this is what I’ve got up so far.
I’m learning a lot, and there is quite a bit to be gained from playing around with something as simple as this. Â My end goal for this is to bring back some of the simple demos from the genesis of this blog, and build more of a directed walk through of some of the principles and features of the system. Â Up to this point, the demos have left it completely up to the user to figure out what to do with them. Â Ideally I’d like it to be so inviting and obvious that viewers just give it a go, but that’s probably too ambitious for the topics.
At work I strive to build systems that leave the simple stuff simple and make the complicated stuff manageable. Â Trying to eliminate the complexity from something that is naturally complicated is just going to end up covering it up and frustrating the user.
This article reminded me of one of the strange features of the standard wave equation from basic physics. Â The wave equation describes an infinitely long vibrating string. Â Like point masses from Newtonian physics each bit of string has a position and velocity. Â If you know the position and velocity of the entire string at one moment of time, you can predict the position and momentum of any bit of string at any time in the past or present. Â It seems natural to think in those terms because we predict the motion of things around us all the time.
The reason this works is that the wave equation lets us solveÂ for the time evolution of the system based on the current state. Â For the case of a string, the time and position variables areÂ symmetric, and knowing the position and velocity of the string at one point for all time is just as good as knowing about all space at one time. Â Philosophically this is a nice feature. Â It means that nothing in this world can escape you, all you have to do is sit in one place and everything either has passed you by or will pass you by, nothing can avoid you forever.
In higher dimensions things are similar, but just a little different. Â For a 3 dimensional space time (2 spacial dimensions and 1 time) knowing the position and velocity for all space at an instant of time is what you need to make predictions. Â Instead of sitting in 1 place and being able to keep track of everything, you need to monitor a single line that divides the world in half. Â Knowing everything that crosses that line for all time lets you know everything in that space. Â In our 3 dimensional spacial world, you have to monitor an entire infinite plane.
That works because signals on this string travel in one direction with one speed, they never turn around or stop. Â That is completely unlike particle dynamics where things move independently. Â This would be just anÂ interesting feature of these wave equations, but basic quantum mechanicsÂ Â is a wave theory, and the same rules apply.
In my last few posts, I’ve been trying to characterize different potentials through the shapes of their orbits (gravitational, harmonic, lorentz). In the middle of it, I came across a post on bad astronomy about gravitational slingshots. I figured it would be a perfect opportunity to use these images to show the effect in a different way.
To start, I’m sending a beam of particles directly towards the planet in question, just like my previous demos. This time I slowly increase the speed of the planet and the orbit shapes change accordingly. It was pretty hard to see what was going on with the first few renderings, so I started by increasing the intensity and color of the orbit with the speed of the particle. Things were starting to look better, but they didn’t really start to illuminate the dynamics until I subtracted off the initial velocity of the test particles and only showed particles that were moving faster than their initial speed.
This one shows the default case of a stationary target. The particles accelerate as they get close and then slow down to their initial speed as they move a way.
This is the first shot of a moving target. You can see there are now a few particles that are being shot back at a higher speed, extending past the first line of acceleration from the last image.
There are a few more features visible now as the speed picks up. Since these are lorentz potentials, you can see a more pronounced central core of paths that pass right by the target and are only slightly deflected. The same thing happens with particles further out, but there is a sweet spot that generates two beams. The faster the target moves, the less deflection is seen. With a slow target, the final trajectory is almost a 180, but the angle decreases more and more as the target speed increases, though the final boost in speed increases as well.
I wanted to finish up with one image that captured the gravitational result. There are many similarities, but some significant differences. The infinity at dead center means that there isn’t a max deflection angle like there is in the lorentz case. As you you approach dead center, there will always be a set of bound states. This leads to simpler images, and really those trajectories aren’t all that interesting as far as the slingshot goes. Hyperbolic orbits are the only ones that get launched somewhere.
Anyway I hope these images illustrate some of the features of this process. I’ve learned a lot putting them together, both about these processes, and how to illustrate them in a way that makes the physics visible to the naked eye.
This is the second in a series profiling the solutions to common central force fields from physics. Check out the the first one on the Newtonian gravitational force field.
These are even more plain than the gravitational well. That’s because these are for the harmonic potential. One of the things that distinguishes this potential is that all orbits are bound and elliptical in shape.
Contrasting that with the gravitational potential, there are 3 orbit shapes for that force: elliptical, parabolic, and hyperbolic. You can see the effects in the image. There is a relatively dark area bounded by the parabolic orbit, with a diffuse bright spot near the tip of the parabola. This is due to the tightly curved hyperbolic orbits nearby. The bright spot fades out into a dim patch further out as the trajectories become straighter the further they are from the center of the field.
Both of these fields have infinities which means that they valid only as approximations to actual forces. The harmonic field has the most severe, it extends out to infinity and no particle can escape from it’s pull. It doesn’t matter how fast our test particles are moving, they will never escape the grasp of the central pull.
Newtonian gravity has the opposite problem. The force becomes infinitely strong as you approach the central point. This causes problems for the differential equation solver and leads to the lower two images with kinked up trajectories and the rays radiating from the central point.
In our next installment I’ll be showing a field without those infinities and see how it stacks up.
I’ve had this sitting on my computer for a while and thought I’d publish it rather then just leave it sitting there. This simulates a pulse of light expanding out like my compression wave examples. This time however, we are looking at a pulse of light expanding in the vicinity of a black hole. Rather than expanding out in a circular wave pulse, it wraps around the hole and circles back to the original location traveling around the hole forever.
These are all images of the same electron orbital. The doughnut or torus shows the electron rotating around the central point in the orbital plane. In the standard terminology this has principal quantum number 2, which represents the energy level of the electron.
The azimuthal quantum number is 1. This represents how much of the energy of the electron is stored in rotating around the atom; in other words, it’s angular momentum.
The magnetic quantum number is 1 as well. This represents an electron spinning as fast as it possibly can given the momentum it has in a circular orbit.
The next image shows a standing wave mode of this orbital. This is where things get kinda weird. This mode is formed by two rotating waves moving in opposite directions. the (2,1,-1) and (2,1,1) modes.
This is where things get interesting. These wavefunctions are complex functions spread out through space. One of the basic assumptions of quantum mechanics is that the square of this wavefunction represents the probability that the particle will be found at that location. The 3 pictures above represent the same probability distributions sampled at different levels of detail.
Because these are probabilities there isn’t any fixed way to map them to a particular shape. The level set for finding an electron within a region 50% of the time can look much different than that for 80% or 99%. For the simple spherical distributions, they do look the same, but for higher energy wavefunctions the shapes get more complicated.
My first attempt to view these functions was a simple addition of probabilities, which produced images like the one to the left. This image corresponds to the 4th image up top. As you can see this looks flat, and you can’t make out the links between the red blobs, the connections between them or the shape of anything. This is a somewhat more accurate rendering of the function, but it lacks all the features like shading, and solid form that our brain uses to give us clues about depth and shape.
On the other hand, the solid rendering hides the blurriness of these methods. Looking at the 3 pictures up top, you can’t tell by just looking at an individual picture how they are connected and what probability cutoff they represent. Looking at just static pictures is even harder.
So now that I’m no longer limited to the static shots on the web, or in a book, I’m seeing how these functions fit together. While the azimuthal part of the wave function is just a scaled cosine function, the interaction with the other dimensions of the wave function produce two different kinds of nodes. The first are the blue nodes that keep the simple blob shape even at very low thresholds. The other is the red nodes that have a dimple in them when they try to connect across the center of the torus at low thresholds. None of this helps me with calculations, but I feel like I have a much clearer view of what’s going on.
None of these pictures holds the whole story. Each one just shows a different part of the puzzle, and I’m sure there are many other ways to highlight many of the other features of these functions.